Removability of sets for sub-polyharmonic functions

33 (2003), 31–42

Removability of sets for sub-polyharmonic functions

Toshihide Futamura, Kyoko Kishi and Yoshihiro Mizuta

(Received January 22, 2002) (Revised April 25, 2002)

Abstract. Our first aim in this paper is to generalize Boˆcher’s theorem for functionsu whose Riesz measurem¼D^muis nonnegative in the punctured unit ballB0. In fact, if usatisfies a certain integral condition andm¼D^mub0 inB0, then it is shown thatu can be written as the sum of a generalized potential ofmand a polyharmonic function on B. This is nothing but the Laurent series expansion for u.

The next aim is to give a polyharmonic version of the recent results by Riihentaus [11] concerning removability of sets for subharmonic functions.

1. Introduction and statement of results

Let Rⁿ be the n-dimensional Euclidean space with a point x¼ ðx1;x₂;. . .;x_nÞ. For a multi-index l¼ ðl1;l₂;. . .;l_nÞ, we set

jlj ¼l₁þl₂þ þl_n; x^l¼x₁^l1x₂^l2. . .x_n^ln and

D^l¼ q qx1

l1 q qx2

l2

. . . q

qxn

ln

:

We denote byBðx;rÞ the open ball centered at x with radius r>0, whose boundary is written as Sðx;rÞ ¼qBðx;rÞ. We also denote by B the unit ball Bð0;1Þ and by B₀ the punctured unit ball B f0g.

A real-valued function u on an open set GHRⁿ is called polyharmonic of order m on G if uAC^2mðGÞ and D^mu¼0 on G, where m is a positive integer, D denotes the Laplacian and D^mu¼D^m1ðDuÞ (cf. [2], [10]). We denote by H^mðGÞ the space of polyharmonic functions of order m on G. In particular, u is harmonic on G if uAH¹ðGÞ.

The fundamental solution of D^m is written as R_2m, that is,

R2mðxÞ ¼ a_mjxj^2mn if 2mn is not an even nonnegative integer, amjxj^2mn logð1=jxjÞ if 2mn is an even nonnegative integer, (

2000 Mathematics Subject Classification. Primary 31B30

Key words and phrases. polyharmonic functions, isolated singularities, Boˆcher’s theorem, Laurent series expansion, Riesz decomposition, removability of sets

where the constant am is chosen so that D^mR2m is the Dirac measure d at the origin. We denote by R_2m;L the remainder term of Taylor expansion of R_2m:

R_2m;Lðz;xÞ ¼R_2mðzxÞ X

jljaL

z^l

l!ðD^lR_2mÞðxÞ for a nonnegative integer L.

We say that a locally integrable function u on an open set GHRⁿ is sub- polyharmonic of order m in G if D^mub0 in G in the weak sense, that is,

ð

G

uðxÞD^mjðxÞdxb0 for all nonnegative jAC^y₀ ðGÞ:

Our first aim in this note is to establish Boˆcher’s theorem for sub- polyharmonic functions uAL_loc¹ ð2B0Þ, where 2B0¼Bð0;2Þ f0g; for poly- harmonic functions, we refer the reader to the previous paper [3] as a gen- eralization of Armitage [1].

Theorem 1. Suppose that uAL¹_locð2B0Þ and m¼D^mu is a nonnegative measure on 2B₀. If u satisfies

ð

2B0

juðxÞj jxj^sdx<y ð1Þ

for some number sbmaxf2m;ng, then uðxÞ ¼

ð

B₀

R2m;Lðz;xÞdmðzÞ þhðxÞ þ X

jljaL

CðlÞD^lR2mðxÞ ð2Þ for a.e. xAB₀, where L is the integer such that sþ2m1<Lasþ2m, hAH^mðBÞ and CðlÞ denote constants.

The above expression is called the Laurent series expansion for u.

To prove Theorem 1, we first show that the generalized potential Ð

B₀R2m;Lðz;xÞdmðzÞ satisfies condition (1) for s⁰>s, and then apply Boˆcher’s theorem for polyharmonic functions on B₀ given in [3].

Next we discuss removability of sets for sub-polyharmonic functions inRⁿ. We say that a continuous function h on ½0;yÞ is a measure function if hð0Þ ¼0, h is nondecreasing and

hð2rÞaMhðrÞ for all r>0; ð3Þ where M is a positive constant. For e>0 and EHRⁿ, write

E_e¼ fxARⁿ:dðx;EÞ<eg;

where dðx;EÞ denotes the distance of x from E, that is, dðx;EÞ ¼ inffjxyj: yAEg. Then the upper Minkowski h-content of E is defined by

M_hðEÞ ¼lim sup

e!0þ

jEej hðeÞ;

where jFj denotes the n-dimensional Lebesgue measure of a set F. If hðrÞ ¼r^na, 0aa<n, then we write Ma for Mh.

We introduce the result by Riihentaus [11] (see also Gardiner [4]).

Theorem A (Riihentaus). Let aA½0;n2 and let E be a closed set in W such that MaðEÞ ¼0. If f is subharmonic in WnE and satisfies

fðxÞadðx;EÞ^aþ2n for all xAWnE;

then f has a subharmonic extension to W.

Now we state the following theorem.

Theorem 2. Let h be a measure function. Suppose E is a closed set in W such that MhðEÞ ¼0. If uAL_loc¹ ðWnEÞis sub-polyharmonic of order m in WnE and satisfies

juðxÞjadðx;EÞ^2mhðdðx;EÞÞ¹ for all xAWnE; ð4Þ then u has a sub-polyharmonic extension to W of order m.

Let h and k be two measure functions on ½0;yÞ such that limr!0

kðrÞ hðrÞ¼0:

In Theorem 2, if MkðEÞ<y and

juðxÞjadðx;EÞ^2mhðdðx;EÞÞ¹ for all xAWnE; ð5Þ then u is shown to have a sub-polyharmonic extension to W (see also Rii- hentaus [11, Theorem 2]).

2. Lemmas

Throughout this paper, let M denote various constants, not neccessarily the same on any two occurrences.

We need several lemmas to prove Theorem 1.

Lemma 1. If u and m are as in Theorem 1, then ð

AðrÞ

dmðzÞaMr^2m ð

CðrÞ

juðzÞjdz

whenever 0<r<¹₂, where AðrÞ ¼ frajxj<2rg and CðrÞ ¼ fr=2<jxj<4rg.

Proof. Consider a function cAC₀^yðCð1ÞÞ such that cb0 and cðxÞ ¼ 1 if 1ajxja2;

0 if jxja1=2 orjxjb4:

If we set c_rðxÞ ¼c ^x_r for 0<r<1=2, then ð

AðrÞ

dmðzÞa ð

CðrÞ

c_rdmðzÞ

¼ ð

CðrÞ

ðD^mc_rÞu dz

a ð

CðrÞ

jD^mc_rj jujdz

aMr^2m ð

CðrÞ

jujdz:

This proves Lemma 1.

Lemma 2. If u and m are as above, then ð

B₀

jzj^ldmðzÞ<y ð6Þ

whenever lbsþ2m.

Proof. Let Aj¼Að2^jÞ and Cj ¼Cð2^jÞ; then we have by Lemma 1 ð

B₀

jzj^ldmðzÞ ¼X^y

j¼1

ð

Aj

jzj^ldmðzÞ

aX^y

j¼1

2^lðjþ1Þ ð

Aj

dmðzÞ

aMX^y

j¼1

2^{lðjþ1Þþ2mj} ð

Cj

juðzÞjdz

aMX^y

j¼1

ð

Cj

juðzÞj jzj^sdz

aM ð

2B0

juðzÞj jzj^sdz<y: We put IðxÞ ¼Ð

B₀jR2m;Lðz;xÞjdmðzÞ, where L is the integer such that sþ2m1<Lasþ2m; note here that Lb0 and Lb2mn because sb maxf2m;ng. For xAB₀, consider the sets

E1¼ zAB₀:jzj<jxj 2

;

E2¼ zAB₀:jzxj<jzj 2

;

E3¼B₀ ðE1UE2Þ:

If 2mbn, then we see from [6, Lemma 4.2] and [9, Lemmas 6, 8, 9] that IðxÞaM

ð

E1

jzj^Lþ1jxj^2mnL1dmðzÞ

þM ð

E2

jzj^2mnþ jzxj^2mnlog jzj jzxj

dmðzÞ

þM ð

E3

jzj^Ljxj^2mnLlog4jzj jxj dmðzÞ

¼MfI1ðxÞ þI2ðxÞ þI3ðxÞg;

if 2m<n, then I₂ðxÞ is replaced by I2ðxÞ ¼

ð

E2

jzxj^2mndmðzÞ:

We prove the following lemma.

Lemma 3. If m is a nonnegative measure on B₀ satisfying (6) and s⁰>

sbmaxf2m;ng, then ð

B

IðxÞjxj^s0dx<y:

Proof. We have only to treat s⁰ satisfying s⁰>s and s⁰1<L2m<s⁰:

First, since ð2mnL1þs⁰Þ þn¼s⁰ ðL2mþ1Þ<0, we have ð

B

I₁ðxÞjxj^s0dx¼ ð

B

ð

E1

jzj^Lþ1jxj^2mnL1dmðzÞ

jxj^s0dx

a ð

B₀

jzj^Lþ1 ð

fx:jxjb2jzjg

jxj^2mnL1þs0dx

! dmðzÞ

¼M ð

B0

jzj^2mþs0dmðzÞ<y with the aid of (6).

Next, noting that jzj=2<jxj<2jzj when zAE2, we have ð

B

I2ðxÞjxj^s0dx¼ ð

B

ð

E2

jzj^2mnþ jzxj^2mnlog jzj jzxj

dmðzÞ

jxj^s0dx

a ð

B₀

jzj^2mn ð

fx:jzj=2<jxj<2jzjg

jxj^s0dx

! dmðzÞ

þ ð

B0

ð

fx:jzxjajzj=2g

jzxj^2mnlog jzj

jzxjjxj^s0dx

! dmðzÞ

aM ð

B0

jzj^2mþs0dmðzÞ

þM ð

B₀

jzj^s0 ð

fx:jzxjajzj=2g

jzxj^2mnlog jzj jzxjdx

! dmðzÞ

aM ð

B0

jzj^2mþs0dmðzÞ<y:

Finally, since ð2mnLþs⁰Þ þn¼s⁰ ðL2mÞ>0, we establish ð

B

I₃ðxÞjxj^s0dx¼ ð

B

ð

E3

jzj^Ljxj^2mnLlog4jzj jxj dmðzÞ

jxj^s0dx

a ð

B0

jzj^L ð

fx:jxja2jzjg

jxj^2mnLþs0 log4jzj jxj dx

! dmðzÞ

aM ð

B0

jzj^2mþs0dmðzÞ<y: Thus we have obtained

ð

B

IðxÞjxj^s0dx<y; as required.

Lemma 4. If u and m are as above, then vðxÞ1uðxÞ

ð

B₀

R_2m;Lðz;xÞdmðzÞAH^mðB0Þ with L as before.

Proof. It is su‰cient to show that D^mv¼0 in B₀ in the weak sense.

Let jAC₀^yðB0Þ. In view of Lemma 3, we can apply Fubini’s theorem to obtain

huv;D^mji¼ ð

B0

R_2mðzxÞ X

jljaL

z^l

l!ðD^lR_2mÞðxÞ 0

@

1

AdmðzÞ;D^mj

¼ ð

B0

ð

B0

R2mðzxÞ X

jljaL

z^l

l!ðD^lR2mÞðxÞ 0

@

1

AD^mjðxÞdx 8<

:

9=

;dmðzÞ

¼ ð

B₀

jðzÞ X

jljaL

z^l l!D^ljð0Þ 0

@

1 AdmðzÞ

¼ ð

B0

jðzÞdmðzÞ

¼hu;D^mji;

* +

since j vanishes in a neighborhood of the origin. This proves hv;D^mji¼0;

as required.

3. Proof of Theorem 1

From Lemmas 3 and 4, we see that vAH^mðB0Þ and ð

B0

jvðxÞj jxj^s0dx<y

for all s⁰>s. In view of [3], we can find hAH^mðBÞ and constants CðlÞ for which

vðxÞ ¼hðxÞ þ X

jljaL

CðlÞD^lR2mðxÞ

holds a.e. on B₀, where L is the integer such that sþ2m1<Lasþ2m.

This implies that u is of the form uðxÞ ¼

ð

B0

R2m;Lðz;xÞdmðzÞ þhðxÞ þ X

jljaL

CðlÞD^lR2mðxÞ for a.e. xAB₀, as required.

In case m¼1, our theorem gives the following simple result.

Corollary. If u is a subharmonic function on 2B0 satisfying ð

2B0

u^þðxÞjxj²dx<y; ð7Þ then u can be extended to a subharmonic function on B, where u^þðxÞ ¼ maxfuðxÞ;0g.

Proof. Since u^þ is subharmonic on 2B0 and satisfies (1) withs¼ 2, we can take L¼0 in Theorem 1, and show that u^þ is of the form

u^þðxÞ ¼ ð

B0

R2ðzxÞdmðzÞ þhðxÞ þCR2ðxÞ

for xAB0, where m¼Du^þb0, mðB0Þ<y, h is harmonic in B and C is a constant. In view of (7),

lim inf

r!0 r¹ ð

Sð0;rÞ

u^þðxÞdSðxÞ ¼0:

Moreover, by [8, Theorem 4.3.1] we see easily that limr!0½rkðrÞ¹

ð

Sð0;rÞ

ð

B₀

R2ðzxÞdmðzÞ

dSðxÞ ¼0;

where kðrÞ ¼1 for nb3 and kðrÞ ¼logð1=rÞ for n¼2, which shows that C¼0. Thus u^þ is extended to a subharmonic function on B. Since uau^þ, u is bounded above near the origin, so that u is extended to a subharmonic function on B by [6, Theorem 5.18].

4. Removability of sets

To prove Theorem 2, we need the following lemma, which is a version of partition of unity (cf. [7]).

Lemma 5. Let fBi:i¼1;. . .;Ng, Bi¼Bðxi;r_iÞ, be a finite collection of balls such that f5¹Big is mutually disjoint. Then there is a family of nonnega- tive functions j_iAC₀^yðRⁿÞ with support suppj_iH2B_i such that PN

i¼1j_iðxÞ ¼1 for xA6^N

i¼1Bi. Furthermore, for each multi-index l, there is a constant Cl

such that

jD^lj_iðxÞjaClr^jlj_i for all xARⁿ and i¼1;. . .;N: ð8Þ Proof of Theorem 2. By our assumption that MhðEÞ ¼0, for e>0, there is r₀, 0<r₀<1, such that

jErjaehðrÞ whenever 0arar0: ð9Þ

We first show that ð

ErnE

dðx;EÞ^2mhðdðx;EÞÞ¹dxaMr^2me: ð10Þ If we put K_j¼ fxARⁿjdðx;EÞ<r2^jg, then

E_rnE¼6

y

j¼0

ðKjnKjþ1Þ:

Hence we have by (9) ð

ErnE

dðx;EÞ^2mhðdðx;EÞÞ¹dx¼X^y

j¼0

ð

KjnKjþ1

dðx;EÞ^2mhðdðx;EÞÞ¹dx

aX^y

j¼0

ðr2^jÞ^2mhðr2^ðjþ1ÞÞ¹jKjj

aMr^2meX^y

j¼0

2^2mj

¼Mr^2me: ð11Þ

From (4) and (11) it follows that ð

ErnE

jujdxaMr^2me: ð12Þ

If we set u¼0 on E, then we see that uAL_loc¹ ðWÞ.

Next we show that ð

W

uðxÞD^mjðxÞdxb0 ð13Þ

for nonnegative jAC₀^yðWÞ. We may assume that 0aja1 and jD^ljja1 for every multi-index jlja2m. We put K¼suppjand take r0>0 such that K_r0HW.

Let 0<4r<r0. By a covering lemma, we can find a finite collection of balls B_i¼Bðxi;rÞ such that f5¹B_ig is mutually disjoint and

6

N i¼1

B_iIK:

By re-indexing if necessary, we can find N such that

2BiVE0 q fori¼1;. . .;N;

2BiVE¼q fori¼Nþ1;. . .;N:

Let j_i be as in Lemma 5. Sinceu is sub-polyharmonic of order minWnE, we see that

ð

2Bi

uD^mðjj_iÞdxb0

for i¼Nþ1;. . .;N, so that

ð

W

uD^mjdx¼ ð

W

uD^m j X^N

i¼1

j_i

!

( )

dx

¼X^N

i¼1

ð

2Bi

uD^mðjj_iÞdx

bX^N

i¼1

ð

2Bi

uD^mðjj_iÞdx

bM r^2m

X^N

i¼1

ð

2Bi

jujdx with the aid of (8). Thus by (12) we have

ð

uD^mjdxbMe;

which gives (13). Consequently, u is sub-polyharmonic of order m in W.

For a measure function h and f AL_loc¹ ðWÞ, define Af;hðxÞ ¼sup

B

r^2mhðrÞ¹ inf

v

ð

B

jfðyÞ vðyÞjdy;

where the supremum is taken over all balls B¼Bðx;rÞHWand the infimum is taken over allvAL¹_locðWÞsuch thatD^mvb0 onB. Further consider the setSf

of all xAW such that lim sup

r!0

r^n2m ð

Bðx;rÞ

jfðyÞ vðyÞjdy>0

for all functions vAL_loc¹ ðWÞ satisfying D^mvb0 on a neighborhood of x.

As in [7] we can prove the following theorem, which gives an extension of a theorem in Gardiner [4].

Theorem 3. If Au;hAL^yðWÞ and HhðSuÞ ¼0, then u has a sub- polyharmonic extension to W, where H_h denotes the Hausdor¤ measure with a measure function h.

5. Remarks on Theorem 1

Suppose thatuAL¹_locð2B0Þ andm¼D^mu is a nonnegative measure on 2B0. Then, as in the book of Hayman-Kennedy [6], u can be represented as

uðxÞ ¼ ð

B₀

R_2m;LðjzjÞðz;xÞdmðzÞ þhðxÞ þX

l

CðlÞD^lR_2mðxÞ ð14Þ for a.e. xAB₀, where LðrÞ is a nonincreasing positive function on ð0;1, hAH^mðBÞ and CðlÞ denote constants. To prove this, we use the estimate

jR2m;lðz;xÞjaM^ljzj^lþ1jxj^2mnl1

whenever 2jzjajxj and 2mn<lþ1, where M is a positive constant depending only on m and n.

Thus our theorem gives a condition which assures that L is bounded and the above sum contains only finite terms.

Remark. We do not know whether u has a similar Laurent expansion or not, if we replace condition (1) by

ð

2B0

u^þðxÞjxj^sdx<y:

References

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Japan 38 (1986), 509–513.

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[10] M. Nicolesco, Recherches sur les fonctions polyharmoniques, Ann. Sci. E´ cole Norm Sup.

52 (1935), 183–220.

[11] J. Riihentaus, Removable sets for subharmonic functions, Pacific J. Math. 194 (2000), 199–208.

Toshihide Futamura and Kyoko Kishi Department of Mathematics Graduate School of Science

Hiroshima University Higashi-Hiroshima 739-8526, Japan e-mail: T. Futamura: toshi@mis.hiroshima-u.ac.jp

e-mail: K. Kishi: kyo@mis.hiroshima-u.ac.jp Yoshihiro Mizuta

The Division of Mathematical and Information Sciences Faculty of Integrated Arts and Sciences

Hiroshima University Higashi-Hiroshima 739-8521, Japan e-mail: mizuta@mis.hiroshima-u.ac.jp